The Probability for a Muon Signal in a Cell

The calorimeter segmentation into cells has been chosen such that a MIP particle like a muon deposits enough energy to be above noise cuts. This is true for muons that traverse the entire volume of a cell almost perpendicularly to the uranium plates of the cell, like prompt muons.

A muon that traverses only a part of the cell might not deposit enough energy and so the signal might be suppressed by the noise cuts.

The energy loss of a muon in the calorimeter follows a Landau distribution. This has been shown in many measurements of the calorimeter prototype modules as well as with halo muons in HERA.

In most of those tests though the muon traversed the calorimeter perpendicularly to the uranium plates.

Figure 6.10: For all cells of muon candidates found in the CC event sample, histogram (a) shows the angle between the muon direction and the sampling direction $\theta$ and histogram (b) shows the energy loss $dE/dX$ for these cells, two peaks are visible.

Figure 6.11: For all cells of muon candidates found in the charged current event sample, histogram (a) shows $dE/dX$ for cells with $\theta$ small: ($\theta < 0.1$ or $\theta > 3.04$). The peak at higher $dE/dX$ from figure 6.10 (b) has disappeared. Histogram (b) shows $dE/dX$ for cells with $\theta$ close to : ( $\vert\theta - 1.57\vert < 0.1$). The peak at higher $dE/dX$ from figure 6.10 (b) is more pronounced.

Figure 6.12: $dE/dX$ for all cells of muon candidates found in the CC event sample for angles $0.1 < \theta < 1.56$ and $1.58 < \theta < 3.04$ together with the Landau distribution that has been fitted to the data (a), and the integral of the Landau distribution (b) which gives the probability for a signal in a cell with an energy cut $E_{cut}$ if the muon traverses $l$ material.

It is possible to show this distribution also for muons traversing the detector at different angles. In figure 6.10 the opening angle between the muon trajectory and the direction perpendicular to the uranium plates (the sampling direction) is shown. This histogram contains data from muon candidates found in the charged current event sample.

Clearly three peaks can be identified. At , and . They correspond to cells coming from halo muon events. For those events the cells in BCAL are penetrated from the side while for FCAL and RCAL the muon hits the cells in sampling direction.

In figure 6.10 the energy deposited in a cell is divided by the length of the muon trajectory through active material. Clearly two peaks can be identified. A cut on the angle under which the cell is traversed shows the reason for those two peaks. In figure 6.11 only cells with the muon hitting the cells almost perpendicularly is shown. The secondary peak has completely disappeared. In figure 6.11 only cells where the muon traverses from the side is shown. Here the secondary peak is very much enhanced.

The second peak stems from muons that only traverse scintillator material. The energy loss is smaller but more light is produced.

In figure 6.12 (a) cells with angles $0.1 < \theta < 1.56$ or $1.58 < \theta < 3.04$ are selected. In the same figure a Landau distribution is shown that has been fitted to the experimental data:

To fit a Landau distribution the following formula is used with and as free parameters:

(38)

The following parameterization results:

(39)

(40)

This parameterization is used for the calculation of the probability to cause a signal. The reason is that MUFFIN has no problem detecting halo muons, muons in general if they run along a module. That is a very clear pattern. More difficult are muons that traverse the detector at other angles. And for those muons the ideal parameterization for the Landau distribution has to be found.

From the $dE/dX$ distribution it is possible to calculate the probability for a muon to deposit so much energy in a cell that the measured cell energy is bigger than the noise cut. For a known length $l$ of the muon trajectory in the cell and a cell cut $E_{cut}$ the probability is simply the integral of the Landau distribution from to infinity:

(41)

is a normalization factor such that : The probability to measure an energy bigger than the $E_{cut}$ is 1.0 if either the energy cut is zero or the amount of material traversed infinite.

The probability not to measure a signal is:

(42)

The probability distribution for the parameterization chosen is shown in figure 6.12 (b): Clearly for a given energy cut a muon has to traverse a certain amount of UCAL material to deposit enough energy to cause a signal above the cut value. The more material is traversed, the higher the probability to cause a signal.

These results are used in the minimization line fit (see section 6.8.2).